\magnification=\magstep2


$$
 L = \mu v r
$$

$$\eqalign {
 L &= (\mu) (v_a) (r_a) \cr
   &= (\mu) (v_a) (a[1 + e]) \cr
}$$

$$
 v_a = \sqrt{ { {GM}\over{a} } { {(1-e)}\over{(1+e)}} }
$$

$$
  L  = \mu \sqrt{ GMa(1 - e^2) }
$$

$$\eqalign{
  {\rm E} &= {\rm KE} + {\rm GPE} \cr
          &= {{1}\over{2}}\mu v^2 + ( - {{GM\mu}\over{r}} )  \cr
} $$


\vfill\eject

$$
  {\rm E} = {{1}\over{2}}\mu {{GM}\over{a}}{{(1-e)}\over{(1+e)}}
                        - {{GM\mu}\over{a(1+e)}}   
$$
 
$$
  {\rm E}  = {{1}\over{2}}  {( - {{G M \mu}\over{a}} )}    
$$

$$
  {\rm E}  = {{1}\over{2}} \thinspace\thinspace\thinspace {\rm GPE_{avg}} 
$$


$$
  {\rm E}  =  {\rm KE } + {\rm GPE}  =  
         {{1}\over{2}} \thinspace\thinspace\thinspace {\rm GPE_{avg}} 
$$

$$
  {\rm E}  =  {\rm KE_{avg} } + {\rm GPE_{avg}}  =  
         {{1}\over{2}} \thinspace\thinspace\thinspace {\rm GPE_{avg}}
$$
          
$$
  {\rm KE_{avg} } = - {{1}\over{2}} \thinspace\thinspace\thinspace {\rm GPE_{avg}}
$$

\vfill\eject

$$
  {\rm KE } = - {{1}\over{2}} \thinspace\thinspace {\rm GPE}
$$

$$
  {{1}\over{2}} \mu v^2 = {{G M \mu}\over{2a}}
$$

$$
  {{1}\over{2}} \mu {\left({{2 \pi a}\over{P}}\right)}^2 = {{G M \mu}\over{2a}}
$$

$$
  P^2  =  {{4 \pi^2}\over{G M }} a^3
$$







\bye
